How this instrument works
A central angle sits at a circle's center, its two arms running out along radii to the ends of an arc. Its size is fixed by one ratio: θ = s ⁄ r, the arc length divided by the radius. That ratio is not a formula bolted onto radians after the fact — it is the definition of the radian. One radian is set, by construction, to be the angle at which the arc exactly equals the radius; stretch the arc to twice the radius and the angle is two radians, with no conversion constant anywhere in sight.
Because the relation is a plain ratio, it scales without a ceiling. An arc equal to the full circumference, 2πr, subtends the full turn, 2π radians — 360°, one complete revolution — for a coin or a crater rim alike. Push the arc past the circumference and the angle keeps climbing past 2π: the two radii sweep around the center more than once. This sheet reports that raw figure rather than folding it back to a first lap, so a central angle here can legitimately read larger than 360°.
This instrument is the mirror of the arc length sheet elsewhere on this site. That one solves s = θr for the arc, given an angle; this one solves the same equation for θ, given the arc. They are one relationship read in opposite directions, and the detail that trips people up is units — the ratio s ⁄ r only ever yields radians directly, so a reading in degrees or turns is a conversion this sheet performs, never a second formula.
- Enter the arc's measured length into Arc length, in any distance unit you choose.
- Enter the circle's Radius in that same unit — the ratio only means something when both share one unit.
- Read Central angle for the result; the underlying math runs in radians, so use the unit control beside the field to view it as deg, rad, or turn.
- Try the defaults first: Arc length 3.14159265 with Radius 2 returns exactly 90°, a quarter turn.
- To check a full lap, set Arc length to 2π times Radius and confirm Central angle lands on 360° (or 2π rad, or 1 turn).
Worked example — a quarter turn from an arc of π
Set Arc length to 3.141592653589793 — that is π — and Radius to 2. The sheet divides: θ = 3.141592653589793 ⁄ 2 = 1.5707963267948966 radians, the figure this instrument holds internally before any unit is chosen. Flip the output to degrees and the same angle reads 90° exactly; flip it to turns and it reads 0.25.
The geometry backs the arithmetic up independently. This circle's full circumference is 2π × 2 = 4π, about 12.566, and the arc entered, π, is exactly one quarter of that distance. A quarter of the circumference has to subtend a quarter of the full turn — 360° ⁄ 4 = 90°, or equivalently 2π ⁄ 4 = π ⁄ 2 radians — which is precisely the 1.5707963267948966 the ratio returns, down to the last bit of precision.
Questions
Why does the central angle formula only work in radians?
Because s ⁄ r is not merely proportional to the radian measure — it is the radian measure, by definition. One radian is the angle where arc length equals radius. Degrees and turns are relabelings applied after that division, using 360° ⁄ 2π and 1 turn ⁄ 2π as conversion factors; the raw ratio itself never produces them directly.
How does the central angle relate to arc length?
They are inverse questions about the same equation, θ = s ⁄ r. Fix the radius and the angle and you can solve for the arc instead: s = θr. This sheet answers 'what angle does this arc make?'; a companion arc length sheet answers 'how long is the arc for this angle?' — same circle, same relationship, opposite unknown.
What happens if the arc length is longer than the circle's circumference?
The angle passes 360° (2π radians, 1 turn) and keeps counting rather than resetting to zero. An arc of one and a half circumferences returns 540°, correctly describing two radii that have swept one and a half times around the center. This is intentional — wrapping the value would hide genuine multi-turn arcs, like a rope wound twice around a drum.
Is a central angle the same as an inscribed angle for the same arc?
No, and they differ by a fixed factor. A central angle has its vertex at the circle's center; an inscribed angle has its vertex on the circle itself, looking at the same arc from the rim. The inscribed angle theorem states the inscribed angle is always exactly half the central angle subtending that arc — a distinct relationship this sheet does not compute.
What is the most common mistake when using this formula?
Mixing units between Arc length and Radius — entering one in centimeters and the other in inches produces a ratio that is neither angle nor anything else meaningful. The formula is unit-agnostic only in the sense that any single length unit works, provided both fields use the identical one throughout the calculation.